Cell Lineage Statistics with Incomplete Population Trees

Cell lineage statistics is a powerful tool for inferring cellular parameters, such as division rate, death rate, or population growth rate. Yet, in practice such an analysis suffers from a basic problem: how should we treat incomplete lineages that do not survive until the end of the experiment? Here, we develop a model-independent theoretical framework to address this issue. We show how to quantify fitness landscape, survivor bias, and selection for arbitrary cell traits from cell lineage statistics in the presence of death, and we test this method using an experimental data set in which a cell population is exposed to a drug that kills a large fraction of the population. This analysis reveals that failing to properly account for dead lineages can lead to misleading fitness estimations. For simple trait dynamics, we prove and illustrate numerically that the fitness landscape and the survivor bias can in addition be used for the nonparametric estimation of the division and death rates, using only lineage histories. Our framework provides universal bounds on the population growth rate, and a fluctuation-response relation that quantifies the change in population growth rate due to the variability in death rate. Further, in the context of cell size control, we obtain generalizations of Powell's relation that link the distributions of generation times with the population growth rate, and we show that the survivor bias can sometimes conceal the adder property, namely the constant increment of volume between birth and division.

Figure 1

Population trees without and with death. (a) Population tree with no death starting with $N_0=1$ cell and ending with $N\left(t\right)=10$ lineages at time $t$. In the table, the columns indicate the number $K$ of divisions along the lineage and the backward and forward weights. (b) Population tree starting with $N_0=1$ cell and ending with $n(\sigma =0,t)=3$ dead lineages (red crosses), and $N\left(t\right)=n(\sigma =1,t)=6$ alive lineages at time $t$. In the table, the survival status $\sigma$ takes value 1 for surviving lineages and 0 for dead lineages. The forward probability of survival is computed as $p_{\text{for}}(\sigma =1,t)=1\times 1/4+3\times 1/8+2\times 1/32=11/16$.

Figure 2

Cartoons of two experimental setups where lineages can end before the end of the experiment. The cell colors represent cell heterogeneity and the white cells are dead. (a) Free growth in bulk; cells may be far from each other or in contact as in a tissue. (b) Cytometer setup from [7] where cells grow in a chamber open at both ends, and they are evacuated by the flow of growth medium in order to maintain the population constant inside the chamber.

Figure 3

Analysis of a lineage tree with biological death. (a) Example of lineage tree of individual bacterial cells proliferating and being killed by drug exposure [11]. The population is exposed to INH on a microfluidic device from $t=0$ h, as indicated by the dashed line. Inset: evolution of the corresponding number $N_t$ of alive cells with time. (b), (d) Distributions and landscapes of the preexposure growth rate (computed 1 h before drug exposure). There are no values at the rightmost bin for $h_t^{★}, h_t$, and $h_t^{†}$ because there are no alive lineages with this value of preexposure growth rate ($p_{\mathrm{for}}^{★}=p_{\mathrm{back}}=0$). (c),(e) Distributions and landscapes of the logarithmic cell size at the end of the lineage (computed at $t=36$ h for surviving lineages or at the time of death for lineages that died before).

Figure 4

Inference of division and death rates from fitness landscape and survivor bias of time-averaged trait in the numerical simulation. (a) A schematic of the population model used in the simulation. A cell with the trait $s$ divides with the probability $r\left(s\right)dt$ and dies with the probability $\gamma \left(s\right)dt$ during $dt$. (b) Filled curve: Stationary forward probability distribution of the trait $s$ when $\gamma \left(s\right)=0$ (no bias). Blue solid and red dashed curves are the division rate $r\left(s\right)$ and death rate $\gamma \left(s\right)$, given in Appendix pp1-s3. (c) Fitness landscape $h_t^{*}\left(\overline{s}\right)$ and survivor bias $h_t^{†}\left(\overline{s}\right)$ of time-averaged trait $\overline{s}$. We choose the parameters so that the population size decreases like what happens in the experimental data we analyzed in Fig. 3, and we choose $t=4$, which is shorter than the autocorrelation time of $s$ and much shorter than the extinction timescale. The points and the error bars represent means and standard deviations over 100 runs, each starting with $N_0=1000$ cells.

Figure 5

Analysis of data from [7] for a population maintained constant thanks to dilution. (a) Illustration of Eq. (35), which shows that the bounds for ${\mathrm{\Lambda }}_t-{\mathrm{\Gamma }}_t$ get tighter as time increases. (b) Separate time evolutions of ${\mathrm{\Lambda }}_t$ and ${\mathrm{\Gamma }}_t$, which show the expected convergence of ${\mathrm{\Lambda }}_t$ toward 0, due to the fact that the population is maintained constant, and a slower convergence of ${\mathrm{\Gamma }}_t$ toward a steady value, not fully achieved within the reach of the experiment.

Figure 6

Influence of autocorrelation factor $r_g$ from simulations on the inference method. Left column: forward distributions (each histogram is evaluated from all 100 runs), right column: comparison between the division rate $r\left(\overline{s}\right)$ [death rate $\gamma \left(\overline{s}\right)$] and the fitness landscape $h_t^{★}\left(\overline{s}\right)$ [survivor bias $h_t^{†}\left(\overline{s}\right)$]. (a),(b) Large autocorrelation $r_g=0.8$. (c)–(f) Small autocorrelation $r_g=0.2$. (a)–(d) Death rate given by $\gamma \left(s\right)=1.2s^{10}/(s^{10}+0.9^{10})$. (e),(f) Flat death rate $\gamma \left(s\right)=0.6$. (a)–(f) Division rate $r\left(s\right)=0.12+0.96s^{10}/(s^{10}+1.1^{10})$ and observation time $t=4$.

Figure 7

Landscapes of (a) preexposure elongation rate and (b) final logarithmic cell size, where the curves and the shaded areas show the mean and $\text{mean}\pm \text{standard}$ deviation of each landscape over 10 000 random shuffles of the trait values. See Appendix pp2 for the details on the procedure for random shuffling of trait values.